Question

Get the Wien displacement law : u( $\nu$ ,T) = $\nu$ ^3 F( $\nu$ /T).

Answer 1

We have to show that = u (v, T) = v^3 F(v/T) where F(v/T) is the function of V and T and V is frequency, and T is then Number of mode from frequency Range v to v + dv is N_v dv = e pi V c^3 dv and energy density = Nv times energy in one node rightarrow u_lambda d lambda = e pi hc/lambda^5 exp (-hc/lambda K_T T) d lambda rightarrow v = c/lambda, dv = -c/lambda^2 d lambda \r\n u_v dv = -e pi h c/(c/v)^5 exp (-hv/k_0 t) dv lambda^2/c u_v dv = e pi h c v^5 dv c^2/c^5 c lambda^2 exp (-h v/k T) u_v dv = e pi h v^3/c^3 dv exp (- hv/k_0 t) u_v dv = v^3 F (v/T) where F (v/T) = e pi h/c^3 exp (-hv/k) T) dv